The ubiquitous $\zeta$-function and some of its "usual" and "unusual" meromorphic properties

TL;DR

This paper classifies the meromorphic structures of zeta functions related to conic manifolds, revealing exotic phenomena such as logarithmic branch cuts and poles with large multiplicities, along with formulas for singularity coefficients.

Contribution

It provides a complete classification and uncovers new exotic meromorphic phenomena of zeta functions associated with conic manifolds.

Findings

Countably many logarithmic branch cuts on the nonpositive real axis

Unusual poles with arbitrarily large multiplicities

Explicit algebraic-combinatorial formulas for singularity coefficients

Abstract

In this contribution we announce a complete classification and new exotic phenomena of the meromorphic structure of $\z$-functions associated to conic manifolds proved in \cite{KLP1}. In particular, we show that the meromorphic extensions of these $\z$-functions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multiplicity. Moreover, we give a precise algebraic-combinatorial formula to compute the coefficients of the leading order terms of the singularities.